Dsp notation

Introduction

Within the Connexions Project , MathML content has been chosen to render mathematical expressions. Although mostcommon expressions can be rendered using MathML, it is known that not all expressions are supported. New notations areconstantly being developed and various academic communities often use specialized notations. The evolution of MathML isan understandably slow process. Because of the diversity of languages and academic cultures, it is impossible to encodeall of the possible notations or even the bulk of mathematical notation in MathML content. However, seeing the necessity forextensibility, the MathML recommendation allows for extensions to be written for MathML using the csymbol tag. A list of extensions written by the Connexions Project can befound on Mountainbunker .

Xsl transformations and macros

Although certain types of content may be supported by MathML, authors may wish to a notational rendering different from the default withoutaltering the meaning of the mathematics. XSL transformations can be used to bind different notation to MathML content tags without altering their semantics.

Signal processing notation

The Connexions project is not yet able to support author written XSLT stylesheets, and the rendering of differentnotations is not yet possible. Therefore, a recommended notation is presented which is to be used until on-the-fly rendering of authorstylesheets is supported by Connexions software. Examples of MathML code are included for each notation to help the use thecorrect notation and MathML code.

Below is a quick-reference table of the notation we recommend for the content whose correct notation is often found writtenwith various variables and symbols. Click on any of the content names to be taking to a brief discussion of it and example code of how to mark-up the MathML correctly.

A listing of all notation presented in this module.

• Time
• Frequency
• Signal
• Special signals
  • Impulse
  • Unit Step
  • Sinc
• Norm
• Convolution
  • Circular Convolution
• Twiddle Factor
• Fourier Transform
  • CTFT
• Laplace Transform
• Z-Transform
• Complex Numbers
• Inner Product

Time

It is suggested that $t$ be used as the continuous variable of time and that $n$ be used as the discrete variable of time.

Period

For discrete-time, periodic functions, we recommend that a capital T, $T$ , be used to denote the period, or sampling period, of a signal.

Frequency

Like time, frequency can be represented as a continuous or as a discrete variable. It is recommended that $f$ Hz be used for the continuous variable of frequency and that $v$ cycles/sample be used as a discrete variable. Further, frequency may be furtherlimited in the discrete case to taking a limited number of values (as in the case of the discrete Fourier transform).In these cases, it is recommended that $k$ is used. Corresponding to the continuous variable, an appropriateradian measure would be $\Omega =2\times \pi f$ rad/sec and for the discrete case $\omega =2\times \pi v$ rad/sample. These last two notations are extremely important as they are seen in most Fourier Transforms andother signal and processing equations. See the code below for example MathML mark-up.